Opening hours vary throughout the year (some places are closed outside the high season). In general hours tend to be longer from June to August, and shorter from September to May. Standard opening hours:
Banks 9am–4pm Monday to Friday
Cafe-bars 10am–1am Sunday to Thursday, 10am to between 3am and 6am Friday and Saturday
Offices 9am–5pm Monday to Friday
Petrol stations 8am–10pm or 11pm (automated pumps open 24 hours)
Post offices 9am–4pm or 4.30pm Monday to Friday (to 6pm in larger towns)
Restaurants 11.30am–2.30pm and 6pm–9pm or 10pm
Shops 10am–6pm Monday to Friday, 10am–4pm Saturday; some Sunday opening in Reykjavík malls and major shopping strips
Vínbúðin (government-run alcohol stores) Variable; many outside Reykjavík only open for a couple of hours per day
Some regional attractions and tourist-oriented businesses in Iceland are only open for a short summer season, typically from June to August. Reykjavík attractions and businesses generally run year-round.
As tourism is growing at a rapid pace, some regional businesses are vague about opening and closing dates; increasingly, seasonal restaurants or guesthouses may open sometime in May, or even April, and stay open until the end of September or into October if demand warrants it.
With the growth of winter tourism, an increasing number of businesses (especially on the Ring Road) are feeling their way towards year-round trading. Note that many Icelandic hotels and guesthouses close from Christmas Eve to New Year’s Day.
Check websites and/or Facebook pages of businesses, and ask around for advice.
Note that many museums outside the capital only have regular, listed opening hours during summer (June to August). From September to May they may advertise restricted opening hours (eg a couple of hours once a week), but many places are happy to open for individuals on request, with a little forewarning – make contact via museum websites or local tourist offices.